On solenoids and their complements

Abstract: 

The problem of classifying solenoids, their complements in S^3, and continuous maps on them was posed by Borsuk and Eilenberg in 1936. At the time, it stimulated substantial advances in algebraic topology and cohomology theory, including Eilenberg’s obstruction theory, Steenrod duality, and the Eilenberg—Mac Lane universal coefficient theorem. In this talk, I will give a historical overview of the problem, and touch upon some new developments obtained with methods from descriptive set theory.